Localization on D - brane and Gauge theory / Matrix model

We consider the effective topological field theory on Euclidean D-strings wrapping on a 2-cycle in the internal space. We evaluate the vev of an appropriate operator corresponding to the chemical potential of D-instantons bounded to the D-strings, and find that it reduces to the partition function of generalized two-dimensional Yang-Mills theory as a result of localization. We argue that the partition function gives a ground canonical ensemble for multi-instanton corrections for 4D N=2 gauge theory in a suitable large N limit. We find two-dimensional gauge theories that provide the instanton partition function for four-dimensional N = 2 theories with the hypermultiplets in the adjoint and fundamental representations. We also propose a partition function that gives the instanton contributions to four-dimensional N = 2 quiver gauge theory. We discuss the relation between Nekrasov’s instanton partition function and the Dijkgraaf-Vafa theory in terms of large N phase transitions of the generalized two-dimensional Yang-Mills theory. [email protected] [email protected]